1. Field of the Invention
The present invention generally relates to computer implemented methods for pricing derivative securities (for example, options) in the finance industry and, more particularly, to such methods having improved efficiency and that select an importance sampling (IS) distribution and combining the selected IS distribution with stratified sampling or Quasi-Monte Carlo (QMC) in novel ways to price financial instruments.
2. Background Description
Monte Carlo simulation is widely used in the finance industry to price derivative securities. However, the method can be quite inefficient because of large variances associated with the estimates. Variance reduction techniques are therefore required. While a large number of such techniques have been developed, more efficient methods are needed for a variety of financial instruments.
A basic survey on general variance reduction techniques, including both the techniques of importance sampling (IS) and stratified sampling, is found in Monte Carlo Methods by J. Hammersley and D. Handscomb, Methuen and Co. Ltd., London (1964), pp. 55-61. A survey on the use of Monte Carlo methods in finance is described by P. Boyle, M. Broadie, and P. Glasserman in xe2x80x9cSimulation Methods for Security Pricingxe2x80x9d, J. Economic Dynamics and Control, Vol. 21, pp. 1267-1321 (1998). A survey and description of the state-of-the-art for variance reduction in finance applications is included in the article entitled xe2x80x9cAsymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Optionsxe2x80x9d by P. Glasserman, P. Heidelberger and P. Shahabuddin. IBM Research Report RC 21178, Yorktown Heights, N.Y. (1998). The use of Quasi-Monte Carlo (QMC) sequences (see, for example, H. Niederreiter, xe2x80x9cRandom Number Generation and Quasi-Monte Carlo Methodsxe2x80x9d, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (1992)) as a variance reduction technique in finance applications has also been considered (see, for example, P. Acworth, M. Broadie and P. Glasserman, xe2x80x9cA Comparison of Some Monte Carlo and Quasi Monte Carlo Techniques for Option Pricingxe2x80x9d, in Monte Carlo and Quasi-Monte Carlo Methods 1996, Lecture Notes in Statistics, Vol. 127, Sringer-Verlag, pp. 1-18 (1998), and W. J. Morokoff and R. Caflisch, xe2x80x9cQuasi Monte Carlo Simulation of Random Walks in Financexe2x80x9d, in Monte Carlo and Quasi-Monte Carlo Methods 1996, Lecture Notes in Statistics, Vol. 127, Sringer-Verlag, pp. 340-352 (1998), and the references therein). The effectiveness of quasi-Monte Carlo (QMC) sequences decreases as the dimension of the problem increases. Therefore, as described in the above references, it is important to assign the lowest dimensions of the QMC sequence to the most xe2x80x9cimportantxe2x80x9d dimensions, or directions.
As described in the above references, the prior art identifies a zero variance estimator; however, it is not practical to implement since it typically requires both knowing the option""s price in advance and sampling from non-standard distributions. In the less general setting of estimating the probability of an event, the prior art also identifies an IS distribution by maximizing a xe2x80x9crate functionxe2x80x9d over the event.
It is therefore an object of the invention to provide a computer implemented method for pricing derivative securities, e.g., options, that selects an importance sampling (IS) distribution combined with stratified sampling or quasi-Monte Carlo (QMC) sequences.
According to the invention, there is provided a process with which securities may be priced. This process consists of the steps of choosing an importance sampling distribution and combining the chosen importance sampling with stratification or quasi-Monte Carlo (QMC) simulation. In the first step, an importance sampling distribution is chosen. In the second step, the chosen importance sampling is combined with stratification or Quasi-Monte Carlo sequencing.
The present invention improves upon earlier methods by selecting an importance sampling distribution in a general, novel and effective way. Furthermore, it also combines importance sampling with stratified sampling in a general, novel and effective way. The pricing of many types of securities reduces to one of estimating an expectation of a real-valued function of some random variables.